To analyze this exponential function $Q = \frac{420}{7 \cdot 10^t}$ and identify the starting value $a$, growth factor $b$, and the percentage growth rate $r$, let's break down the expression.

1. Starting Value $a$: The starting value $a$ is the value of $Q$ when $t = 0$.

   Substituting $t = 0$ into the equation: $Q = \frac{420}{7 \cdot 10^0} = \frac{420}{7 \cdot 1} = \frac{420}{7} = 60$ So, $a = 60$.

2. Growth Factor $b$: The growth factor $b$ is the factor by which $Q$ changes as $t$ increases by 1. Since the function involves $10^t$ in the denominator, this suggests exponential decay rather than growth. Thus, $b$ would be a fraction less than 1 for decay.

   Observing the denominator, we can see that $Q = a \cdot \left(\frac{1}{10}\right)^t$. This means the growth factor $b = \frac{1}{10}$.

3. Percentage Growth Rate $r$: For exponential functions in the form $Q = a \cdot b^t$, if $b = 1 + r$, then $r$ can be interpreted as the growth rate. Since $b = \frac{1}{10}$, we have: $b = 1 - 0.9$ So, $r = -0.9$ or a decay rate of $90\%$.

Summary:

• Starting Value $a$: 60
• Growth Factor $b$: $\frac{1}{10}$
• Percentage Growth Rate $r$: -90% (indicating a 90% decay per time unit)

Would you like further details on exponential decay functions or how these values are derived?

Here are some related questions for further exploration:

1. How would the function change if it represented exponential growth instead of decay?
2. What is the impact on the function if the growth factor $b$ is changed to 2?
3. Can exponential functions model both growth and decay? How?
4. How does the base of the exponent (e.g., 10 in this case) affect the function's behavior?
5. What real-world scenarios can be modeled by exponential decay functions like this?

Tip: Always remember that in exponential functions, a growth factor $b < 1$ indicates decay, while $b > 1$ indicates growth.